I wanted to share what I wrote in answer to this question. It was part of an assignment for the ProGen study group I'm in. We are studying Tom Jone's new textbook Mastering Genealogical Proof.
I think that it's good to emphasize Family History. Everybody can participate, and their participation is helpful both to current and future generations of Family Historians. If you are LDS, doing Family History is particularly relevant to past generations because it allows you to do temple work for the dead.
One of the goals of the study group is to bring an ancestor's brick wall and try to break through it together. I chose to research my husband's 5th great grandfather, Patrick McQueen. Like many Czech ancestors, he seemed to appear out of nowhere; a village of origin problem. Patrick arrived in Glengarry, Canada sometime in the late 1820's, after which he moved to Westville, Franklin, New York. He and his wife Ann Moran had 12 kids. His son William McQueen moved west and became the Sheriff of Salt Lake City. This was during the period of Mormon polygamy. My understanding is that William was specifically elected because he was Catholic (not Mormon), and that this somehow had to do with polygamy and anti-polygamy goings-on. Anyway, we are looking for his father's village of origin in Ireland. I refer to this in my answer. Just wanted to clarify in case there was confusion.
Back to the question. "How do you distinguish genealogy from family history?"
To me, very roughly speaking, math/science : humanities as genealogy : family history. Math/science is results-oriented. Humanities are process-oriented.
My husband is a bioinformatics scientist. He likes systematic logic. He uses Bayesian statistics on a daily basis to test and prove theories. Brevity and simplicity matter. And if your argument makes sense, he will agree with you (handy!).
As a French Teaching major, I was in the school of Humanities. In my French classes, as in most world language classes, we spent a lot of time practicing communicating. We talked a lot about ourselves and conversed with others in the class. We studied the tools (grammar, vocab) and put them into action by applying them in a self-relating context. Self-expression and communication are more important than brevity, or even consistency in content (what you say, not how you say it - the language rules matter). In my pedagogy classes, we also spent a lot of time reading and discussing methodology and theory, and then practicing the act of teaching (in class, in groups, practicums, and student teaching).
Genealogy is the study of family relationships. It is a rigorous field that relates to hundreds of other disciplines. It demands thorough data gathering, strict source citation, logical analysis of the data, resolution of conflicts, and a written documentation of the results so that others can understand what you have done. The goal of genealogy is to prove relationships. It is a results-oriented field.
Family History is the study of family relationships in a context. It is also a rigorous field, but instead of being results-oriented, it is process-oriented. One goal of Family History may be to gain a relationship with a loved one who has died. In my opinion, the best way to reach this process-oriented goal would be to follow the same GPS model that is (or should be!) the standard in the field of Genealogy: thorough data gathering, strict source citation, logical analysis, conflict resolution, and written documentation. The end goal, however, is not the proof argument. It is the intangible kinship you gain with the person you researched.
Family History can be less intimidating than genealogy because you can start with yourself. You don't have to be an expert in anything except you. Because your goal is different, you can start small. I think every Genealogist probably starts as a Family Historian. To be an excellent Genealogist, you need a solid grounding in genealogy theory, for example: applying the GPS, learning the history of the place in which you are researching, paleography, etc.
Ultimately, Family Historians experience firsthand the flaws in approaching the past without the GPS. They will want to learn more genealogy theory, perhaps become certified by some accrediting body, participate in conferences and local organizations, join some forums and/or progen study groups ;) etc.
You can do Genealogy without doing Family History. When I do work for a client, I am doing Genealogy. My goal is to answer the client's question; it is results-oriented.
You can do Family History without doing Genealogy. When I assemble a scrapbook (okay I never do this - but some of you might!) of photos of my kids, or read my ancestor's diary, I am doing Family History. My goal is to leave documentation of my family's life, or to gain a closeness to my family by reading their's; it is process-oriented.
You can do both Genealogy and Family History at the same time. When I work to break through my brick wall, my goals are both to prove Patrick McQueen's Irish origin, and to gain an understanding, appreciation, and love for him. They are both goal and process-oriented.